To use Ultimate Guitar's app to import into OnSong, open a song in UG's app > Tap the More icon (3 dot icon in the upper right-hand corner) > Share > Open With... > Tap Add to OnSong -- ((If "Add to OnSong" isn't on the main list there, tap the more option - tap the green plus sign by Add to OnSong)) > Switch over to the OnSong app > Import > Duplicate (after the first time). This can only be done one song at a time.
To be transparent, there is currently a bug in Ultimate Guitar's app that causes it to strip the name in the metadata section. So when you import a song into OnSong, every new song imported overrides the last song unless you tap Duplicate and go into the Song Editor to add the metadata information right away.
For example, if song one is Last Christmas, it will import into OnSong, but you'll need to immediately edit it to look like the following:
Then, for the next song that you import, tap Duplicate and go into the Song Editor to edit that song's metadata similarly, and so on...
According to UG's support team, their developers are aware of this bug and are working on fixing it. Unfortunately, there isn't anything that can be done in the meantime.